This post is referenced here: "Applying Artificial Intelligence and Machine Learning to Finance and Technology".

(⚠️ : shameless plug.)

Single line Sudoku solver: https://rosettacode.org/wiki/Sudoku#Mathematica

solve[sudoku_] := 
 NestWhile[
  Join @@ Table[
     Table[ReplacePart[s, #1 -> n], {n, #2}] & @@ 
      First@SortBy[{#, 
           Complement[Range@9, s[[First@#]], s[[;; , Last@#]], 
            Catenate@
             Extract[Partition[s, {3, 3}], Quotient[#, 3, -2]]]} & /@ 
         Position[s, 0, {2}], 
        Length@Last@# &], {s, #}] &, {sudoku}, ! FreeQ[#, 0] &]

I am not familiar with MS-OML but it seems clear enough what this particular nesting does. We'll look at one such.

"Foreach [{i,I},Unequal[Foreach[{j,J},x[i,j]]]]"

Taking this apart, for each i in the "row" subscript set, we set up an unequal involving ALL of {x[i,1],x[i,2],...x[i,9]}.

I don't understand why the Foreach statements are nested in the MS-OML code.

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Difficult for me to comment in the absence of actual code. If it was something like this Reduce[allcons, vars, Integers]; I will agree: exact ILP can be slow. That's why I often write branch-and-prune loops using NMinimize.

Reduce uses exact methods. Which might be somewhat slower than FindMinimum, but still should work for purposes of solving a Sudoku.

See also:

http://library.wolfram.com/infocenter/Conferences/6528/

Page 23 of the nb has an ILP-based sudoku solver.

That's a neat way of doing it! I actually wrote a solved on my last flight (a week ago or so) from USA back to Europe. I was having doubts with the in-flight entertainment system version of Sudoku; it generated non-unique Sudokus...so the puzzles had many many solutions... big shame. I'm still considering sending Delta airlines an email about it...